On the self-decomposability of the Fr\'echet distribution

Let $\{{\mathrm{\Gamma }}_t,t\ge 0\}$ be the Gamma subordinator. Using a moment identification due to Bertoin-Yor (2002), we observe that for every $t>0$ and $\alpha \in (0,1)$ the random variable ${\mathrm{\Gamma }}_t^{-\alpha }$ is distributed as the exponential functional of some spectrally negative L'evy process. This entails that all size-biased samplings of Fr'echet distributions are self-decomposable and that the extreme value distribution $F_{\xi }$ is infinitely divisible if and only if $\xi \in ̸(0,1),$ solving problems raised by Steutel (1973) and Bondesson (1992). We also review different analytical and probabilistic interpretations of the infinite divisibility of ${\mathrm{\Gamma }}_t^{-\alpha }$ for $t,\alpha >0.$